Download 5 The Renormalization Group

g2
I
g4
G
WF
II
Figure 8: The RG flow for a scalar theory in three dimensions, projected to the (g2 , g4 )
subspace. The Wilson–Fisher and Gaussian fixed points are shown. The blue line is the
projection of the critical surface. The arrows point in the direction of RG flow towards the
IR.
ends at the Wilson–Fisher fixed point in the IR. Theories on the line heading vertically out
of the Gaussian fixed–point correspond to massive free theories, while theories in region
I are massless and free in the deep UV, but become interacting and massive in the IR.
These theories are parametrized by the scalar mass and by the strength of the interaction
at any given energy scale. Theories in region II are likewise free and massless in the UV
but interacting in the IR. However, these theories have g2 < 0 so that the mass term is
negative. This implies that the minimum of the potential V (φ) lies away from φ = 0, so
for theories in region II, φ will develop a vacuum expectation value, %φ& =
3 0. The RG
trajectory obtained by deforming the Wilson–Fisher fixed point by a mass term is shown
in red. All couplings in any theory to the right of this line diverge as we try to follow the
RG back to the UV; these theories do not have well–defined continuum limits.
5.3.5
Zamolodchikov’s C–theorem
Polchinski’s equation showed that renormalization group flow could be understood as a
form of heat flow. It’s natural to ask whether, as for usual heat flow, this can be thought
of as a gradient flow so that there is some real positive function C(gi , Λ) that decreases
monotonically along the flow. Notice that this implies C = const. at a fixed point gi∗ , and
that C(gi∗ , Λ) > C(gi∗∗ , Λ$ ) whenever a fixed point gi∗∗ may be reached by perturbing the
theory a fixed point gi∗ by a relevant operator and flowing to the IR. In 1986, Alexander
Zamolodchikov found such a function C for any unitary, Lorentz invariant QFT in two
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